Construction of coupled Harry Dym hierarchy and its solutions from Stäckel systems

Nonlinear Analysis 73 (2010) 3004–3017

Contents lists available at ScienceDirect

Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Construction of coupled Harry Dym hierarchy and its solutions fromStäckel systemsKrzysztof Marciniak a,∗, Maciej Błaszak ba Department of Science and Technology, Campus Norrköping, Linköping University, 601-74 Norrköping, Swedenb Faculty of Physics, A. Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland

a r t i c l e i n f o

Article history:Received 15 March 2010Accepted 25 June 2010

Keywords:Stäckel separable systemsHamilton–Jacobi theoryHydrodynamic systemsRational solutionsMulticomponent Harry Dym hierarchy

a b s t r a c t

In this paper we show how to construct the coupled (multicomponent) Harry Dym (cHD)hierarchy from classical Stäckel separable systems. Both nonlocal and purely differentialparts of hierarchies are obtained. We also construct various classes of solutions of cHDhierarchy from solutions of corresponding Stäckel systems.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Various relations between finite- and infinite-dimensional nonlinear integrable systems have been investigated sincethe middle of 70s in a long sequence of papers starting from paper [1], through papers [2–5] (see for example [6] for moredetailed bibliography) and many others. In all these efforts, however, the main idea was to pass from infinite- to finite-dimensional integrable systems. This paper is a third paper in our series of papers showing that an opposite way is also pos-sible: that of passing from ordinary differential equations integrable in the sense of Arnold–Liouville to infinite-dimensionalintegrable systems (soliton hierarchies). In paper [7] we demonstrated a way of generating commuting evolutionary flowsfrom corresponding family of Stäckel systems (that is classical finite-dimensional Hamiltonian systems quadratic in mo-menta and separable in the sense of Hamilton–Jacobi theory). We presented our idea in the setting of coupled (multicom-ponent) KdVhierarchies (for definition and properties of these hierarchies, see for example [8]). In paper [9]we systematizedand developed this idea by showing how solutions of these Stäckel systems can be used for generating various classes ofsolutions of cKdV hierarchies. Although both papers have been written for the case of cKdV, similar constructions are possi-ble for other hierarchies as well. In this paper we demonstrate a way of generating the coupled (i.e. multicomponent) HarryDym (cHD) hierarchy (see [10,11]) and various classes of its solutions from a class of Stäckel systems of Benenti type. Ourmethod leads both to the nonlocal cHD hierarchy as well as to purely differential cHD hierarchy, that is to amulticomponentgeneralization of HD hierarchy discussed in [12] (see also [13]). The nonlocal part of cHD hierarchy has not been discussedin [10] at all. We also clarify and simplify some of the results given in [7,9].The paper is organized as follows. In Section 2 we briefly remind some basic fact about Stäckel separable systems and

discuss how they are related to corresponding Killing systems (dispersionless nonlinear PDE’s of evolutionary type definedbyKilling tensors of Stäckel systems). Sections 3 and 4 are devoted to the description of nonlocalmulticomponentHarryDymhierarchy and its various solutions, respectively. Sections 5 and 6 are devoted to local (purely differential) cHD hierarchy.

∗ Corresponding author. Tel.: +46 11 363320; fax: +46 11 363270.E-mail addresses: [email protected] (K. Marciniak), [email protected] (M. Błaszak).

0362-546X/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2010.06.067

http://www.elsevier.com/locate/na

http://www.elsevier.com/locate/na

mailto:[email protected]

mailto:[email protected]

http://dx.doi.org/10.1016/j.na.2010.06.067

K. Marciniak, M. Błaszak / Nonlinear Analysis 73 (2010) 3004–3017 3005

2. Stäckel systems and their dispersionless counterpart

Stäckel separable systems can bemost conveniently obtained from an appropriate class of separation relations. Generallyspeaking, n equations of the form

ϕi(λi, µi, a1, . . . , an) = 0, i = 1, . . . , n, ai ∈ R (1)

(each involving only one pair λi, µi of canonical coordinates on a 2n-dimensional PoissonmanifoldM) are called separationrelations [14] provided that det

(∂ϕi∂aj

)6= 0. We can then locally resolve Eq. (1) with respect to ai obtaining

ai = Hi(λ, µ), i = 1, . . . , n (2)

with some new functions (Hamiltonians) Hi(λ, µ) that in turn generate n canonical Hamiltonian systems onM:

λti =∂Hi∂µ

, µti = −∂Hi∂λ, i = 1, . . . , n. (3)

All the flows (3) mutually commute since the Hamiltonians Hi Poisson commute. Moreover, Hamilton–Jacobi equationsfor all the Hamiltonians Hi are separable in the (λ, µ)-variables since they are algebraically equivalent to the separationrelations (1).In this article we consider a special but important class of separation relations, namely

n∑j=1

ajλn−ji = λ

mi µ

2i +

ε

4λki , i = 1, . . . , n (4)

with arbitrary fixed m, k ∈ Z,ε = ±1 (the constant 14 is not essential for the construction and is only introduced for asmoother identification our systems with the hierarchy in [10]). The relations (4) are linear in the coefficients ai so that theycan be (globally) solved by Cramer formulas, which yields

ai = µTKiG(m)µ+ε

4V (k)i ≡ H

n,m,ki , i = 1, . . . , n,m, k ∈ Z (5)

where we denote λ = (λ1, . . . , λn)T and µ = (µ1, . . . , µn)T . Functions Hi defined as the right-hand sides of (5) depend onm and k and can be interpreted as n quadratic in momenta µ Hamiltonians on the phase spaceM = T ∗Q cotangent to aRiemannian manifoldQ parametrized by (λ1, . . . , λn) and equipped with the contravariant metric tensor G(m) (dependingonm ∈ Z) given by:

G(m) = diag(λm1

∆1, . . . ,

λmn

∆n

)with∆i =

∏j6=i

(λi − λj). (6)

It can be shown that G(m) is of zero curvature for m = 0, . . . , n and that G(n+1) is of non-zero constant curvature, whileall other choices of m lead to spaces of non-constant curvature. The Hamiltonians Hn,m,ki are known in literature as StäckelHamiltonians and the corresponding commuting Hamiltonian flows (3) are then called Stäckel systems, or more precisely,Stäckel systems of Benenti type. They are obviously separable in the sense of Hamilton–Jacobi theory since they by the verydefinition satisfy Stäckel relations (4). The objects Ki in (5) are Killing tensors for any metric G(m) and are given by

Ki = −diag(∂qi∂λ1

, . . . ,∂qi∂λn

)i = 1, . . . , n,

where qi = qi(λ) are Viète polynomials (signed symmetric polynomials) in λ:

qi(λ) = (−1)i∑

1≤s1<s2<···<si≤n

λs1 . . . λsi , i = 1, . . . , n (7)

that can also be considered as new coordinates on the Riemannian manifold Q (we will then refer to them as Viètecoordinates). Notice that Ki do not depend on neitherm nor k. Finally, the potentials V

(k)i can be constructed recursively [15]

by

V (k+1)i = V (k)i+1 − qiV(k)1 , k ∈ Z, with V (0)i = δin, (8)

where we put V (k)i = 0 for i < 0 or i > n. The first potentials are trivial: V(k)i = δi,n−k for k = 0, 1, . . . , n − 1. The first

nontrivial potentials are V (n)i = −qi. For k > n the potentials V (k)i become complicated polynomial functions of q. Therecursion (8) can also be reversed

V (k)r = V(k+1)r−1 −

qr−1qnV (k+1)n , k ∈ Z, r = 1, . . . , n, (9)

3006 K. Marciniak, M. Błaszak / Nonlinear Analysis 73 (2010) 3004–3017

leading to potentialsV (k)i with k < 0. These potentials startwithV(−1)i = −

qi−1qnand are rather complicated rational functions

of q. They will be referred to as negative potentials. It can also be shown [7] that

g(m)ij = V(2n−m−i−j)1 (10)

where g(m) =(G(m)

)−1 is the corresponding covariant metric tensor.Remark 1. The general n-time (simultaneous) solution for Hamilton equations (3) associated with all the Hamiltonians (5)is given implicitly by

ti + ci = ±12

n∑r=1

∫λn−ir√√√√λmr

(n∑j=1ajλ

n−jr −

ε4λkr

)dλr , i = 1, . . . , n. (11)

To see this it is enough to integrate the related Hamilton–Jacobi problem. Now, with n Hamiltonians Hn,m,ki in (5) we canassociate, by corresponding Legendre transforms, n Lagrangians Ln,m,ki : TQ→R given by

Ln,m,ki (λ, λti) =14λTtig

(m)K−1i λti −ε

4V (k)i , i = 1, . . . , n. (12)

Every Lagrangian Ln,m,ki give rise to n systems of Euler–Lagrange equations

Esj (Ln,m,ki ) = 0, j = 1, . . . , n (13)

(each for every s between 1 and n) where

Esj =∂

∂λj−ddts

∂

∂(∂λj/∂ts

) , j = 1, . . . , n

are components of the Euler–Lagrange operator with respect to the independent variable ta.

Remark 2. By construction, the solutions (11) are also general solutions for all the Euler–Lagrange equations (13). It meansthat for a particular s the general solution of Euler–Lagrange equations Esj (L

n,m,ki ) = 0 is given by (11) where tp for p 6= s

plays a role of a constant parameter.

Denote now the variable t1 as x (our method works similarly with any ti chosen as x). With every Killing tensor Ki fori = 2, . . . , nwe can associate a dispersionless evolutionary PDE of the form

λti = Kiλx ≡ Zi [λ] i = 2, . . . , n (14)

(where λ = (λ1, . . . λn)T ). We will call PDE’s in (14) simply Killing systems. Here and in what follows we use the notationf [λ] to denote integral–differential function of λ i.e. a function of λ, its x-derivatives and antiderivatives (integrals). Inthe case above Zi [λ] = Zi(λ, λx). The chosen variable t1 = x in (14) plays thus the role of a space variable while theremaining variables ti should then be considered as evolution parameters (times). Eq. (14) constitute a set of n−1 integrabledispersionless equations that due to the form of Ki belong to the class of weakly nonlinear semi-Hamiltonian systems,i.e. hydrodynamic-type systems that are semi-Hamiltonian in the sense of Tsarev [16,17] andweakly nonlinear [18]. Actually,the systems (14) are finite-component restrictions of the universal hydrodynamic hierarchy considered in [19]. The variablesλi are Riemann invariants of all the system (14) as Ki are diagonal in λ. The systems (14) can also be considered as n − 1dynamical systems on some infinite-dimensional function space V of vectors (λ1(x), . . . , λn(x))with Zi being n− 1 vectorfields on V . It can be shown [18] that the vector fields Zi commute on V:[

Zi, Zj]= 0 i, j = 2, . . . , n.

Note also that sinceK1 = Iwecan complete the systemof Eq. (14) by the equationλτ = K1λx = λx ≡ Z1with the translation-invariant general solution λi = λi(x+ τ). The vector field Z1 also commutes with all the vector fields Z2, . . . , Zn [18].

Proposition 3. Every mutual solution λ(t1, . . . , tn) (11) of all Hamiltonian systems (3)with Hamiltonians of Benenti type (5) is(after replacing t1 with x) also a particular solution of all n− 1 corresponding Killing systems in (14).

Proof. Let us assume that a vector function λ(t1, . . . , tn) solves (11). Then, by construction, it also solves the spatial partof (3) with appropriate functions µ(t1, . . . , tn) given by µi = ∂W (λ, a)/∂λi (W = W (λ, a) is a common integral of all theHamilton–Jacobi equations for Hamiltonians Hn,m,ki ). It means that λ(t1, . . . , tn) solves


λti =∂

∂µHn,m,ki = 2KiG(m)µ, i = 1, . . . , n. (15)

Since K1 = I we get from the first equation in (15)µ(t1, . . . , tn) = 12g

(m)(λ(t1, . . . , tn))λt1(t1, . . . , tn). Substituting it to theremaining equations in (15) yields then

λti(t1, . . . , tn) = Ki(λ(t1, . . . , tn))λt1(t1, . . . , tn), i = 2, . . . , n

which concludes the proof as t1 = x. Thus, all the solutions (11) also solve all n− 1 Killing systems in (14). �

Moreover, we have

Theorem 4. The general (n-time) solution of all the Killing systems in (14) is given by

ti + ci =n∑r=1

∫λn−ir

ϕr(λr)dλr , i = 1, . . . , n (16)

(where ϕr are arbitrary functions of one variable).

The proof of this statement can be found in [18]. Obviously, (16) contains all the solutions (11).Suppose now that a particular solution (16) of our Killing systems (14) is of themore specific form (11). Since this class of

solutions – by construction – satisfies all the Euler–Lagrange equations (13), we can treat equations (13) as additional bondsthat these solutions satisfy. We can therefore use these bonds to express some variables λi by other λ’s. Thus, within theclass (11) of solutions (16) of Killing systems (14) we can perform a variable elimination (reparametrization) that turns (14)into entirely new sets of evolutionary PDE’s. As we have demonstrated in [7,9], in carefully chosen cases and in a particularcoordinate system (Viète coordinates (7)) this reparametrization turns systems (14) into systems with dispersion (solitonhierarchies) with the solution (11) being also a solution of these new systems with dispersion. In this paper wewill produceby this method (the local and the nonlocal part of) the coupled (multicomponent) Harry Dym hierarchy.

3. Nonlocal coupled Harry Dym hierarchy

Assume now that ε = 1 in (4) and therefore also in (11), (12) etc. In order to perform the elimination procedure justmentioned, let us pass to Viète coordinates as given in (7). The Killing systems (14) are tensorial so in Viète coordinates theyhave the form

qti = Ki(q)qx, i = 2, . . . , n

or, explicitly

ddtiqj = (qj+i−1)x +

j−1∑k=1

(qk(qj+i−k−1

)x − qj+i−k−1 (qk)x

)≡(Zni [q]

)j, j = 1, . . . , n (17)

(where we put qα = 0 for α > n), where i = 2, . . . , n and where(Zni [q]

)j denotes the jth component of the vector fieldZi [q]. The superscript n at Zi indicates the number of components in the vector field Zi and we will sometimes use it sincewe will need to switch between various n. From (17) one can see that

(Zni [q]

)j=(Znj [q]

)i for all i, j = 1, . . . , n. Obviously,G(m), g(m) and Ki are tensors and can thus also easily be transformed to Viète coordinates.Consider now Euler–Lagrange equations (13) with s = 1 (so that ts = t1 = x) associated with Lagrangians L

n,m,k1 denoted

further on for simplicity as Ln,m,k. Denote also E1i as Ei, i = 1, . . . , n and consider the equations

Ei(Ln,m,k

)= 0, i = 1, . . . , n, n,∈ N, k ∈ Z, (18)

written in q-variables, so that now

Ei =∂

∂qi−ddx

∂

∂qi,x, i = 1, . . . , n,

while (since K1 = I)

Ln,m,k = Ln,m,k(q, qx) =14qTx g

(m)qx −14V (k)1 . (19)

As it has been shown in [7] the following symmetry relations are satisfied for α = 1, . . . , n− 1

Ei(Ln,m,k

)= Ei−α

(Ln,m+α,k−α

), i = α + 1, . . . , n, (20)

that can also be written as

Ei(Ln,m,k

)= Ei+α

(Ln,m−α,k+α

), i = 1, . . . , n− α. (21)


Due to (20) and (21) the Eqs. (18) can be embedded in the following double-infinite multi-Lagrangian ‘‘ladder’’ ofEuler–Lagrange equations of the form

E1(Ln,m+j−1,k−j+1

)= E2

(Ln,m+j−2,k−j+2

)= · · · = En(Ln,m+j−n,k−j+n) = 0, j = . . . ,−1, 0, 1, . . . (22)

with fixedm, k ∈ Z (the Eqs. (18) fit in (22) at j = 1, 2, . . . , n). For a given dimension n the ladder (22) is determined by thesumm+ k in the sense that various choices ofm and kwith the samem+ k yield the same ladder.We are now ready to present our elimination procedure leading to multicomponent integral (nonlocal) Harry Dym

hierarchy. Assume thatwewant to produce first s−1 flows of theN-component (N ∈ N) hierarchy. Let us take n = s+N−1,m = −N and k = 0 in (12), that is, let us consider the purely kinetic Lagrangian Ln,−N,0 with n = s + N − 1 and thecorresponding Euler–Lagrange equations (18). Due to this special choice of all parameters the last n − N equations in (18)attain the form

EN+1(Ln,−N,0

)≡ −

12qn,xx + ϕn−N [q1, . . . , qn−1] = 0,

EN+2(Ln,−N,0

)≡ −

12qn−1,xx + ϕn−N−1[q1, . . . , qn−2] = 0,

...

En−1(Ln,−N,0

)≡ −

12qN+2,xx + ϕ2[q1, . . . , qN+1] = 0

En(Ln,−N,0

)≡ −

12qN+1,xx + ϕ1[q1, . . . , qN ] = 0

(23)

and are a part of the ladder (22) withm+ k = −N . Now, by direct calculation of Ei(Ln,−N,0

)with the use of some identities

satisfied by the potentials V (i)1 it can be proved that

EN(Ln,−N,0

)= EN+1

(Ln+1,−N,0

)+12qn+1,xx,

Ei(Ln,−N,0

)= Ei+1

(Ln+1,−N,0

), i = N + 1, . . . , n.

These identities lead to

Proposition 5. The functions ϕi in (23) do not depend on n in the sense that increasing n to n + 1 (and keeping N constant)turn (23) into n− N + 1 equations

EN+1(Ln+1,−N,0

)= −

12qn+1,xx + EN

(Ln,−N,0

)≡ −

12qn+1,xx + ϕn−N+1[q1, . . . , qn] = 0,

EN+2(Ln+1,−N,0

)= EN+1

(Ln,−N,0

)≡ −

12qn,xx + ϕn−N [q1, . . . , qn−1] = 0,

......

...

En(Ln+1,−N,0

)= En−1

(Ln,−N,0

)≡ −

12qN+2,xx + ϕ2[q1, . . . , qN+1] = 0,

En+1(Ln+1,−N,0

)= En

(Ln,−N,0

)≡ −

12qN+1,xx + ϕ1[q1, . . . , qN ] = 0.

(24)

It means that increasing n to n + 1 (and keeping N constant) in (23) does not alter these equations except that a newequation of the form

EN+1(Ln+1,−N,0

)≡ −

12qn+1,xx + ϕn−N+1[q1, . . . , qn] = 0

is added at the top of (23). As we will see soon, this will result in the fact that our construction indeed yields an infinitehierarchy of commuting flows.Due to their structure, Eqs. (23) can be formally solved with respect to the variables qN+1, . . . , qn, which yields

qN+1, . . . , qn as some nonlocal (integral–differential) functions of q1, . . . , qN :

qN+1 = f1 [q1, . . . , qN ]...qn = fn−N+1 [q1, . . . , qN ] ,

(25)

where, due to Proposition 5, the functions fi do not depend on n, so increasing n by 1 (and keepingN constant)will only resultin one new equation at the bottom place in (25). Let us now replace the variables qN+1, . . . , qn in the first N components ofthe first s − 1 Killing systems (17) by the corresponding functions fi (right-hand sides of (25)). This yields equations of theform


qtr = ZNr [q] r = 2, . . . s (26)

where q denotes the first N entries in q i.e. q = (q1, . . . , qN)T . They are in general highly nonlinear autonomous systems ofN evolution equations for q1, . . . , qN .

Theorem 6. The vector fields ZNr [q] in (26) do not depend on s in the sense that if we increase s by one in our procedurethen (26) are unaltered and a new equation qts+1 = Z

Ns+1 [q] appears.

Proof. This theorem is a consequence of Proposition 5. If we increase s to s + 1 and keep N constant we have to taken + 1 instead of n in our procedure as n = s + N − 1. Due to (17) we have

(Zn+1r

)j=(Znr)j for r = 2, . . . , s and for

j = 1, . . . ,N i.e. the first N components of the first s− 1 of Killing systems (17) do not change when we increase n to n+ 1.Moreover, as we explained above, the n− N functions fi in (25) do not change either. So, the elimination procedure for thefirst s − 1 vector fields Zi is not altered leading to exactly the same vector fields Z

Nr [q] with r = 2, . . . , s while the vector

field Zn+1s+1 yields the vector field ZNs+1 [q] i.e. a new equation at the end of the sequence (26). �

Repeating this argument we can increase s indefinitely. Thus, our procedure leads to an infinite hierarchy of evolutionaryvector fields (flows)

qtr = ZNr [q] r = 2, 3, . . . (27)

in the sense that if we wish to produce any first s − 1 flows (26) of the hierarchy we can perform our procedure withn = s+N − 1. This way we can obtain arbitrary long sequences of the same infinite set of vector fields with dispersion thatpairwise commute (soliton hierarchy):

Theorem 7. The vector fields ZNr [q] commute i.e.[ZNi , Z

Nj

]= 0 for any i, j = 2, 3, . . . .

This theorem is due to the fact that the original vector fields Zni commute and that the Euler–Lagrange equationsEi(L(n,m,k)) = 0 are invariant with respect to all the fields Zni [7]. Moreover, the vector fields Z

Ni still commute with

ZN1 =

(q1,x, . . . , qN,x

)T . As we demonstrate below, the hierarchy (27) is the nonlocal part of the multicomponent HarryDym soliton hierarchy as discussed in [12].

Example 8. Consider first N = 1 (one-component hierarchy as discussed in [12]). Suppose that we want to obtain the firsts− 1 = 2 flows of the hierarchy. We have then to take n = s+ N − 1 = 3 and consider the elimination equations (23) forthese parameters. The pure kinetic Lagrangian L3,−1,0 has the form

L3,−1,0 =12q21,x

(q1q2 −

12q3 −

12q31

)+12q1,xq2,x

(q21 − q2

)−14q22,xq1 −

12q1,xq3,xq1 +

12q2,xq3,x

so that (13) become

E2(L3,−1,0

)≡ −

12q3,xx +

12q1,xxq2 +

12q2,xxq1 −

12q1,xxq21 −

12q1q21,x +

12q1,xq2,x = 0, (28)

E3(L3,−1,0

)≡ −

12q2,xx +

12q1,xxq1 +

14q21,x = 0.

Due to their specific structure, we can solve (28) with respect to q2 and q3. We will thus use (28) to eliminate variables inthe corresponding n = 3-component Killing systems (17) that have in this case the form:

ddt2

(q1q2q3

)=

( q2,xq3,x + q1q2,x − q2q1,xq1q3,x − q3q1,x

)= Z32 ,

ddt3

(q1q2q3

)=

( q3,xq1q3,x − q3q1,xq2q3,x − q3q2,x

)= Z33 . (29)

By the second equation in (28) we obtain

q2,xx =12q21,x + q1,xxq1.

Integrating it once we obtain

q2,x =12q1q1,x +

12∂−1q1q1,xx


where

∂−1 =

∫. . . dx+ ϕ(t2, t3)

is the integration operator with the integration parameter ϕ that has to be chosen from case to case and has therefore to betreated as a part of the solution of every integration problem. It is always possible to find such a function. Integrating q2,xwe obtain

q2 =14q21 +

12∂−2q1q1,xx.

Further, the first equation in (28) yields

q3,xx = q1,xxq2 + q2,xxq1 − q1,xxq21 − q1q21,x + q1,xq2,x. (30)

Inserting to it q2 and q2,x, as calculated above, and integrating once we obtain

q3,x = −12∂−1q21q1,xx +

12q1∂−1q1q1,xx +

14q21q1,x +

12q1,x∂−2q1q1,xx. (31)

By inserting the obtained formulas for q2,x and q3,x into the first N = 1 components of Z2 and Z3 we obtain the first twoflows of our nonlocal soliton hierarchy:

q1,t2 =12q1q1,x +

12∂−1q1q1,xx = Z2, (32)

q1,t3 = −12∂−1q21q1,xx +

12q1∂−1q1q1,xx +

14q21q1,x +

12q1,x∂−2q1q1,xx = Z3.

Observe that in this particular case we did not have to calculate q3 since it does not enter into the first component of neitherZ2 nor Z3. We needed however q2 in order to calculate q3,x. The flows (32) commute due to Theorem 7.

Example 9. Let us now take N = 2 and s − 1 = 1 so that n = 3 again. We will thus eliminate n − N = 1 variables(namely q3) from the first N = 2 components of the field Z32 above. The elimination equations (13) reduce now toE3(L3,−2,0

)= 0. But, according to (20), E3

(L3,−2,0

)= E2

(L3,−1,−1

)= E2

(L3,−1,0

), the last equality due to the fact that

L3,−1,−1 = L3,−1,0− 14V

(−1)1 = L3,−1,0+ 1

4q3. Thus, the elimination equation E3

(L3,−2,0

)= 0 coincides with the first equation

in (28) and yields exactly (30). Plugging its integrated form (31) into the first two components of Z32 yields the first flow ofthe two-component nonlocal cHD hierarchy:

ddt2

(q1q2

)=

(q2,x

q1q2,x − q2q1,x −12∂−1q21q1,xx +

12q1∂−1q1q1,xx +

14q21q1,x +

12q1,x∂−2q1q1,xx

)= Z2. (33)

The map

ui = EN−i+1(LN,0,0

), i = 1, . . . ,N (34)

transforms the hierarchy (27) into the nonlocal part of the coupled Harry Dym hierarchy (see [10] for its local part) that isthe generalization of the one-field nonlocal HD hierarchy presented in [12]. For example, for N = 2 this map reads

u1 = −12q1,xx

u2 = −12q2,xx +

14q21,x +

12q1q1,xx

and applied to the field Z2 above yields

ddt2

(u1u2

)= −2

(u1,x∂−2u1 + 2u1∂−1u1 −

12u2,x

u2,x∂−2u1 + 2u2∂−1u1

).

4. Solutions of the multicomponent nonlocal HD hierarchy

We will now construct a variety of solutions of the hierarchy (27).


Theorem 10. For any β ∈ {0, . . . , n− 1} the functions λi = λi(t1, . . . , tn) given implicitly by

ti + ci = ±12

n∑r=1

∫λn−ir√√√√λ−N+βr

(n∑j=1ajλ

n−jr −

14λ−βr

)dλr , i = 1, . . . , n (35)

are such that the corresponding functions qi = qi(x = t1, t2, . . . , tn), i = 1, . . . ,N, given by (7) are solutions of the first n− β(n − 1 for β = 0, 1) equations of the N-component integral cHD hierarchy (27). The variables t2, . . . , tn−β+1 (t2, . . . , tn forβ = 0, 1) play then the role of evolution parameters (dynamical times) while the remaining ti’s are free parameters.

For the proof of this theorem, see Appendix. We will now consider some particular, interesting classes of solutions (35).Assume that β = 0 in (35) and that aj = 1

4δj,n + δj,n−γ for some γ ∈ {0, . . . , n− 1}. Then (35) attain the form

ti + ci = ±12

n∑r=1

∫λn−ir√λ−N+γr

dλr , i = 1, . . . , n,

that integrated yields

ti + ci = ±1

2 (n− i+ N/2− γ /2+ 1)

n∑r=1

λn−i+N/2−γ /2+1r , i = 1, . . . , n. (36)

The above system can be algebraically solved with respect to λi only for two choices of γ , namely γ = N and γ = N+1, butit turns out that the case γ = N leads to trivial solutions (polynomial solutions not depending on x). Thus, we must assumeγ = N + 1. In this case the above equations attain the form

ti + ci = ±1

2 (n− i+ 1/2)

n∑r=1

λn−i+1/2r , i = 1, . . . , n. (37)

Note that (37) do not depend on N . It means that for any N between 1 and n − 2 (as γ = N + 1 ≤ n − 1) the functionsq1(x, t2, . . . , tn), . . . , qN(x, t2, . . . , tn) obtained from (37) through (7) solve the first n − 1 equations in (27). The followingtwo examples illustrate this.

Example 11. Assume that n = 3. Then (37) attain the form (with x = t1, ci = 0, we also choose only+ in (37))

x =15

3∑i=1

z5i =15

(ρ51 − 5(ρ1ρ2 − ρ3)(ρ

21 − ρ2)

)t2 =

13

3∑i=1

z3i =13

(ρ31 − 3ρ1ρ2 + 3ρ3

)(38)

t3 =3∑i=1

zi = ρ1

where zi = λ1/2i , i = 1, 2, 3 and where ρ1 =

∑3i=1 zi, ρ2 = z1z2 + z1z3 + z2z3 and ρ3 = z1z2z3 are elementary symmetric

polynomials in zi. The right-hand sides of (38) follow from Newton formulas:n∑i=1

zmi =∑

α1+2α2+···+nαn=m

(−1)a2+α4+α6+···m(α1 + α2 + · · · + αn − 1)!

α1! . . . αn!ρα11 ρ

α22 . . . ραnn form < n, (39)

expressing sums of powers of variables as functions of their symmetric polynomials (these formulas can easily be extendedto the case m ≥ n by taking n′ = m and putting all ρn+1, . . . , ρm equal to zero). The system (38) can be solved explicitlyyielding the solution (37) in ρ-variables:

ρ1 = t3

ρ2 =−15x− 2t53 + 15t

23 t2

5(3t2 − t33

) (40)

ρ3 =15t2t33 + 45t

22 − t

63 − 45xt3

15(3t2 − t33

) .

On the other hand, according with (7) and with (39)

q1 = −(λ1 + λ2 + λ3) = −(z21 + z

22 + z

23

)= −(2ρ2 − ρ21 ).


Plugging (40) into the above identity we obtain

q1(x, t2, t3) = q1(ρi(x, t2, t3)) =t53 + 15t

33 t2 − 30x

5(3t2 − t33

) . (41)

According to Theorem 10, the function q1(x, t2, t3) given by (41) yield a two-time solution to the first n − 1 = 2 flows ofthe nonlocal 1-field (i.e. with N = 1) HD hierarchy (27), i.e. to both systems (32) (after an appropriate choice of integrationconstants).

Example 12. Let us now take n = 4. In this case the Eqs. (37) read (again wit all ci = 0 and with+ only and due to (39))

x =17

4∑i=1

z7i =17

(ρ71 − 7(ρ1ρ2 − ρ3)

((ρ21 − ρ2)

2+ ρ1ρ3

)− 7ρ4(ρ31 − 2ρ1ρ2 + ρ3)

)t2 =

15

4∑i=1

z5i =15

(ρ51 − 5(ρ1ρ2 − ρ3)(ρ

21 − ρ2)− 5ρ1ρ4

)(42)

t3 =13

4∑i=1

z3i =13

(ρ31 − 3ρ1ρ2 + 3ρ3

)t4 =

4∑i=1

zi = ρ1

where as before zi = λ1/2i and ρi are again symmetric polynomials of the variables z1, . . . , z4. This system can again be

algebraically solved forρ1, . . . , ρ4 although the solutions are too complicated to present themhere.We have now, accordingwith (7),

q1 = −(λ1 + λ2 + λ3 + λ4) = −(z21 + z

22 + z

23 + z

24

)= −(2ρ2 − ρ21 )

q2 = λ1λ2 + · · · + λ3λ4 = z21z22 + · · · + z

23z24 = ρ

22 − 2ρ1ρ3 + 2ρ4.

Substituting the variables ρi obtained by solving (42) into these expressions we obtain expressions for q1(x, t2, t3, t4) andq1(x, t2, t3, t4):

q1(x, t2, t3, t4) =P1(x, t2, t3, t4)Q (t2, t3, t4)

, q2(x, t2, t3, t4) =P2(x, t2, t3, t4)Q 2(t2, t3, t4)

(43)

where Pi and Q are rather complicated, but perfectly manageable for any computer algebra program, polynomials. Morespecifically

P1(x, t2, t3, t4) = −17

(105t34 t2 − t

84 − 21t

54 t3 + 630t2t3 − 630xt4 + 315t

23 t24

)and

Q (t2, t3, t4) = 45t2t4 + t64 − 15t3t34 − 45t

23 ,

while P2 is a quadratic in x polynomial that is too complicated to present it here. Now, according to Theorem 10 and thetheory above, the function q1(x, t2, t3, t4) in (43) solves the first n−1 = 3 1-field flows of the hierarchy (27), so in particularboth the flows (32), while the vector function(

q1(x, t2, t3, t4)q1(x, t2, t3, t4)

)solves the first n− 1 = 3 flows of the N = 2-field cHD hierarchy (27) starting with (33).Let us also remark that formulas (36) often lead to implicit solutions of (27). We illustrate it in the following example.

Choose N = 1, n = 2 and γ = 0 in (36). This yields (again for ci = 0)

x =15

(z51 + z

52

), t2 =

15

(z31 + z

32

)(44)

(with zi = λ1/2i ) that cannot be algebraically solved. However, (44) can be embedded in the algebraically solvable system

(38) in the sense that (38) reduces to (44) if we put z3 = 0 or equivalently ρ3 = 0, since ρ3 = z1z2z3. By virtue of Theorem 10it means that the function

q1(x, t2, y(x, t2)) =y(x, t2)5 + 15y(x, t2)3t2 − 30x

5(3t2 − y(x, t2)3

)with the variable y(x, t2) defined implicitly by the equation


15t2y3 + 45t22 − y6− 45xy = 0

(i.e. by the last equation in (40) with y instead of t3), also satisfies the first flow of the nonlocal HD hierarchy i.e. the firstflow in (32).

5. Differential (local) cHD hierarchy and its solutions

We will now obtain the purely differential part of cHD hierarchy as well as a class of its implicit solutions. We choosenow ε = −1 in (4) in order to obtain real solutions in the local case (note that it does not influence the potentials V (m)r ).Analogously to the case of nonlocal hierarchy, wewill perform some variable elimination on the sequence of Killing systems(17). Suppose thus that we want to produce the first s flows of the N-component local (i.e. purely differential) Harry Dymhierarchy. Put n = s+ N and consider the first n− N Euler–Lagrange equations for the Lagrangian Ln,n−N,−n. Using the factthat V (−j)1 = V (−j)1 (qn−j+1, . . . , qn) it can be shown that they attain the form

E1(Ln,n−N,−n

)≡14q2n+ γ

(N)1 [q1, . . . , qN ] = 0, (45)

Ei(Ln,n−N,−n

)≡ −

qn−i+12q3n

+ γ(N)i,1 [q1, . . . , qN−i+1]+

1

qi+1nγ(N)i,2 [qn−i+2, . . . , qn] = 0, i = 2, . . . , n− N

where as usual qα = 0 for α < 1. Note that (45) and (23) belong to the same ladder (22) of Euler–Lagrange equations sincein both casesm+ k = −N .

Proposition 13. The functions γ (N)i,1 , γ(N)i,2 , γ

(N)1 do not depend on n in the sense that increasing n to n+1will not alter (45) except

that a new equation originates at the bottom of the list (45).

The proof of this proposition resembles the proof of the analogous statement for nonlocal case i.e. Proposition 5. Notenow that the structure of (45)makes it possible to eliminate (express) the variables qN+1, . . . , qn as (purely differential now)functions of q1, . . . , qN (although now, opposite to the nonlocal case, we first calculate qn, then qn−1 and so on up to qN+1):

qn = f(N)1 [q1, . . . , qN ] ,

...

qN+1 = f (N)n [q1, . . . , qN ] .

(46)

Now, let us replace the variables qN+1, . . . , qn in the first N components of the last s systems in (17). That leads to s highlynonlinear (purely differential) evolutionary equations of the form

qtr = ZNr [q] r = n− s+ 1 = N + 1, . . . n (47)

where as before q = (q1, . . . , qN)T but with new, purely differential, vector fields ZNr . These fields constitute in fact the first

s fields of the local cHD hierarchy. Contrary to the nonlocal case, however, the first field of the hierarchy appears as the lastequation in (47) i.e. qtn = Z

Nn [q], the second field is qtn−1 = Z

Nn−1 [q] and so on so that the fields of the hierarchy originate in

(47) in the reverse order. We will therefore introduce a new notation and denote

τp = tn−p+1, and XNp = Z

Nn−p+1, p = 1, . . . , n− 1 (48)

so that qtn = ZNn [q] reads qτ1 = X

N1 [q] and so on. The sequence (47) becomes therefore

qτr = XNr [q] , r = 1, . . . , s. (49)

A theorem analogous to Theorem 6 explains that this procedure leads to a hierarchy.

Theorem 14. The vector fields in (49) do not depend on s in the sense that if we increase s to s + 1 then the above eliminationprocedure produces the same sequence (49) of evolutionary systems plus a new system qτs+1 = X

Ns+1 [q] at the end of the

sequence (49) (i.e. at the beginning of the sequence (47)).

Proof. Consider the s systems (47) and increase s to s+1 keeping N constant. We have then to take n+1 instead of n in ourelimination procedure. Since, according to Proposition 13, the functions γ (N)i,1 , γ

(N)i,2 , γ

(N)1 do not depend on n the functions

f (N)i do not depend on n either. It means that increasing n to n+ 1 (and keeping N constant) turns the Eqs. (46) into

qn+1 = f(N)1 [q1, . . . , qN ]

...

qN+2 = f (N)n [q1, . . . , qN ]qN+1 = f

(N)n+1 [q1, . . . , qN ]


and at the same time the structure of the last s equations in (17) changes so that qn is replaced by qn+1, qn−1 is replaced byqn and so on until qN+2. It means that the last s equations in the (extended to n + 1) sequence (47) will after eliminationremain the same while a new equations originates — this time before (with lowest r) the other s ones. �

Thus, by taking appropriate swe can produce on demand an arbitrary (finite) number of evolutionary vector fields

qτr = XNr [q] , r = 1, 2 . . . (50)

and due to same argument as in the nonlocal case, these vector fields all mutually commute:[XNi , X

Nj

]= 0 for all i, j = 1, 2 . . . .

The described procedure leads in fact to multicomponent local Harry Dym hierarchy.

Example 15. Let us first produce the first s = 2 flows of the standard Harry Dym hierarchy i.e. with N = 1. We haven = s+ N = 3. Consider the Lagrangian

Ln,n−N,−n = L3,2,−3 =14q21,x −

q2,xq3,x2q3

+q23,xq24q23+q14q23−q224q33

and the corresponding Euler–Lagrange equations (45). They attain the form

E1(L3,2,−3

)≡14q23−12q1,xx = 0

E2(L3,2,−3

)≡ −

q22q33−q23,x4q23+q3,xx2q3= 0

and can thus easily be solved with respect to q2 and q3 yielding (46) in the explicit form

q3 = q3[q1] =(2q1,xx

)−1/2q2 = q2[q1] =

12

(5q21,xxx − 4q1,xxq1,xxxx

) (2q1,xx

)−7/2.

Substituting these expressions to the first (since N = 1) component of the last s = 2 Killing systems of the sequence (17)we obtain the following two commuting flows:

q1,t2 = (q2[q1])x , q1,t3 = (q3[q1])xor

q1,τ1 = (q3[q1])x = X11, q1,τ2 = (q2[q1])x = X

12

(with the differential functions q2[q1] and q3[q1] given as above) i.e. the first two members of the well known local HarryDym hierarchy.

Example 16. Let us now produce the first s = 2 flows of the N = 2-component Harry Dym hierarchy, we need thereforen = s+ N = 4. The Euler–Lagrange equations (45) for the Lagrangian Ln,n−N,−n = L4,2,−4 attain the form

E1(L4,2,−4

)≡14q24+12q1q1,xx +

14q21,x −

12q2,xx

E2(L4,2,−4

)≡ −

q32q34−12q1,xx

that is soluble with respect to q3 and q4 yielding

q4 = q4[q1,q2] = −w−1/2 ≡ −(2q2,xx − q21,x − 2q1q1,xx

)−1/2q3 = q4[q1,q2] = −q1,xxw−3/2.

Substituting these functions to the first N = 2 components of the last s = 2 Killing systems of the sequence (17) (withn = 4) yields the desired flows

ddτ1

(q1q2

)= X

21 ≡

( (w−1/2

)x

q1(w−1/2

)x − w

−1/2q1,x

)(51)

and

ddτ2

(q1q2

)= X

22 ≡

( (q1,xxw−3/2

)x

q1,xxw−3/2q1,x − q1(q1,xxw−3/2

)x +

(w−1/2

)x

). (52)


Our parametrization of Harry Dym hierarchy differs from the parametrization given in [10]. Generally speaking, thehierarchy (50) is transformed into the multicomponent Harry Dym hierarchy presented in [10] through a complex versionof the map (34)

ur = −iEN−r+1(LN,0,0

), r = 1, . . . ,N, i2 = −1. (53)

For example, in theu-variables the system (51) attains the form

ddτ1

(u1u2

)= X

21 [u] ≡

14

(u−1/22

)xxx

u1(u−1/22

)x+12u−1/22 u1,x

that is exactly the flow (24a) in [10].We will now formulate a theorem corresponding to Theorem 10, i.e. we will generate a wide class of solutions of the

hierarchy (50).

Theorem 17. For any β ∈ {0, . . . , n− 1} the functions λi = λi(t1, . . . , tn) given implicitly by

ti + ci = ±12

n∑r=1

∫λn−ir√√√√λ−N+βr

(n∑j=1ajλ

n−jr +

14λ−βr

)dλr , i = 1, . . . , n (54)

are such that the corresponding functions qi = qi(x = t1, t2, . . . , tn), i = 1, . . . ,N, given by (7) are solutions of the first n− β(n − 1 for β = 0, 1) equations of the N-component integral cHD hierarchy (50). The variables tβ+1 = τn−β , . . . , tn = τ1(t2, . . . , tn for β = 0, 1) are evolution parameters (dynamical times) while the remaining ti’s are free parameters.

Wewill not prove this theorem here as its proof resembles the proof of Theorem 10. Comparing Theorems 10 and 17 wecan see that the solutions (35) and (54) are for β = 1, . . . , n− 1 related through the transformation β → n− β , ε→−ε.i.e. every solution (35) for β = 1, . . . , n − 1 coincides, after changing ε → −ε, with the solution (54) with β ′ = n − β . Italso means that the nonlocal flow qtn−β+1 = Z

Nn−β+1 [q] and the local flow qτβ = X

Nβ [q] share the same family of solutions,

namely (35) (or (54) with β ′ = n− β and with ε′ = −ε).It turns out that (54) cannot be explicitly solved. However, by taking all ai = 0 in (54) (which yields the so called zero-

energy solutions) we can obtain interesting implicit solutions to our hierarchy (50).

Example 18. Consider the solutions (54) with N = 2, n = 3 and with all ai = 0. They have the form

ti + ci = ±3∑r=1

∫λ4−ir dλr , i = 1, 2, 3 (55)

(the same for all β since β-terms cancel after inserting ai = 0) and according to Theorem 17 they solve the first n− 1 = 2flows of the N = 2-component cHD hierarchy (50) i.e. both the flows (51) and (52). Eqs. (55) after integrating yield(remember that t1 = x; we also put all ci = 0 for simplicity of the formulas)

x =14

3∑i=1

λ4i , t2 =13

3∑i=1

λ3i , t3 =12

3∑i=1

λ2i (56)

which cannot be algebraically solved. However, similarly as in the nonlocal case, we can embed (56) in the system

x =14

4∑i=1

λ4i =14

(ρ41 − 4ρ

21ρ2 + 2ρ

22 + 4ρ1ρ3 − 4ρ4

)t2 =

13

4∑i=1

λ3i =13

(ρ31 − 3ρ1ρ2 + 3ρ3

)(57)

t3 =12

4∑i=1

λ2i =12

(ρ21 − 2ρ2

)t4 =

4∑i=1

λi = ρ1

(where ρi are symmetric polynomials in λi so that qi = (−1)i ρi) in the sense that putting λ4 = 0 (so that ρ4 = 0 sinceρ4 = λ1λ2λ3λ4; the right-hand sides of (57) are again due to (39)) in (57) we obtain (56). The Eqs. (57) can be explicitlysolved yielding.


q1 = −ρ1 = −t4

q2 = ρ2 = −t3 +12t24

q3 = −ρ3 = −t2 −16t34 + t3t4

q4 = ρ4 = −x+124t44 −

12t24 t3 +

12t23 + t2t4.

Thus, the functions qi(x, t2, t3) given implicitly by

q1 = −ρ1 = −t4(x, t1, t2)

q2 = ρ2 = −t3(x, t1, t2)+12t4(x, t1, t2)2

where t3(x, t1, t2) and t4(x, t1, t2) are any pair of functions identically satisfying the condition

0 = −x+124t44 −

12t24 t3 +

12t23 + t2t4

solve both (51) and (52).

6. Conclusions

In this article we presented a novel method of obtaining multicomponent Harry Dym hierarchy (both its local andnonlocal part) as well as wide classes of its solutions, from a family of finite-dimensional separable systems (Stäckel systemsof Benenti type). This method has been previously applied to coupled Korteweg–de Vries hierarchy where it produced novelrational solutions and also a family of implicit solutions. In the case of cHD hierarchy discussed here, the method producesamong others rational and implicit solutions in case of nonlocal hierarchy and implicit solutions of the local part. In addition,the method produces wide families of other solutions that are to be exploited elsewhere. It also indicates the existence ofcommon solutions of local and nonlocal cHD systems.Our method can hopefully be extended to other systems, for example by taking more general separation relations than

relations (4).

Acknowledgements

Both authors were partially supported by Swedish Research Council grant no. VR 2009-414 and by Ministry of Scienceand Higher Education (MNiSW) of the Republic of Poland research grant no. N N202 4049 33.

Appendix

We prove here Theorem 10. We start with the case β = 0. For β = 0 the solutions (35) are just solutions (11) with ourchoice ofm and k, namelym = −N, k = 0. The functions

q1(x = t1, t2, . . . , tn), . . . , qn(x = t1, t2, . . . , tn) (58)

obtained from (35) (with β = 0) through (7) satisfy thus all n−1 Killing systems (17). Moreover they satisfy all the Eqs. (23)and thus also all the Eqs. (25) used in our elimination procedure. This means that we are free to use any part of (23) or(25) to perform an elimination of variables in (17). Such elimination thus leads to new equations that are satisfied by thosefunctions from the set (58) that survive the elimination. Now, we know that replacing the variables qN+1, . . . , qn in the firstN components of the first s−1 = n−N Eqs. (17) by the functions given by (25) leads to the first s−1 flows of the hierarchy(27). That means precisely that the first N functions in (58)

q1(x = t1, t2, . . . , tn), . . . , qN(x = t1, t2, . . . , tn) (59)

satisfy the first s − 1 = n − N equations in (27). We will now show that they actually solve the first n − 1 equation in(27). Consider the next flow qts+1 = Z

Ns+1 [q] in (27). In order to obtain this flow, we have to perform the elimination of

variables qN+1, . . . , qn, qn+1 in the flow qts+1 = Zn+1s+1 [q] through (25) written for n+1 instead of n i.e. obtained from solving

(24). This elimination is therefore performed with the help of the same functions qi = qi[q1, . . . , qN ] as for n plus a newfunction qn+1 = fn−N+2 [q1, . . . , qN ]. However,

(Zn+1s+1 [q]

)j=(Zns+1 [q]

)j for all j = 1, . . . ,N − 1 (it follows from (17))while

(Zn+1s+1 [q]

)Ncontains the additional variable qn+1 not present in

(Zns+1 [q]

)N . It means that solutions (59) will certainlysatisfy the first N − 1 components in qts+1 = Z

Ns+1 [q]. Further, since EN+1

(Ln+1,−N,0

)= −

12qn+1,xx + EN

(Ln,−N,0

), the

function qn+1 = fn−N+2 [q1, . . . , qN ] is (after choosing both integration constants equal to zero) identically equal to zero


on the solutions (59). That means that on the solutions (59) we have(Zn+1s+1 [q]

)N=(Zns+1 [q]

)N which means indeed that(59) solves qts+1 = Z

Ns+1 [q]. By expanding this argument, the functions qn+1, qn+2, . . . , qn+N−1 obtained from (23) with n

replaced by n′ = n+ N − 1 i.e. from the n′ − N = n− 1 equations

EN+1(Ln′,−N,0

)≡ −

12qn+N−1,xx + ϕn−1[q1, . . . , qn+N−2] = 0,

EN+2(Ln′,−N,0

)≡ −

12qn+N−2,xx + ϕn−2[q1, . . . , qn+N−3] = 0,

...

En′−1(Ln′,−N,0

)≡ −

12qN+2,xx + ϕ2[q1, . . . , qN+1] = 0

En′(Ln′,−N,0

)≡ −

12qN+1,xx + ϕ1[q1, . . . , qN ] = 0

(60)

(which are necessary to obtain the first n − 1 flows of (27)) are identically zero on the solutions (59) which leads to theconclusion that (59) indeed solve the first n− 1 equations of (27).Assume finally that 0 < β ≤ n − 1. The functions (35) are then the complete solution (as usual, through the map (7))

of all the Euler–Lagrange equations Ei(Ln,β−N,−β) associated with the Lagrangian Ln,β−N,−β . As such, they still must solve allthe Killing systems (17). However, since Ei(Ln,β−N,−β) = Ei+β(Ln,−N,0) for i = 1, . . . , n − β due to (20), for any β > 1 welose the first β − 1 equations in (60) which means that our proof works only for the first n − β flows in (27) — we simplycannot ‘‘blow up’’ n to n′ = n+ N − 1 but only to n′′ = n+ N − 1− β .

References

[1] O.I. Bogojavlenskiı̆, S.P. Novikov, The connection between the Hamiltonian formalisms of stationary and nonstationary problems, Funkcional. Anal. iPriložen. 10 (1) (1976) 9–13 (in Russian). English translation: Functional Anal. Appl. 10 (1) (1976) 8–11.

[2] M. Antonowicz, A.P. Fordy, S. Wojciechowski, Integrable stationary flows: Miura maps and bi-Hamiltonian structures, Phys. Lett. A 124 (1987)143–150.

[3] M. Antonowicz, S. Rauch-Wojciechowski, Restricted flows of solitonhierarchies: coupledKdVandHarryDymcase, J. Phys. A 24 (21) (1991) 5043–5061.[4] S. Rauch-Wojciechowski, K. Marciniak, M. Blaszak, Two Newton decompositions of stationary flows of KdV and Harry Dym hierarchies, Physica A 233(1–2) (1996) 307–330.

[5] K. Marciniak, Coupled Korteweg-de Vries hierarchy with sources and its Newton decomposition, J. Math. Phys. 38 (11) (1997) 5739–5755.[6] M. Błaszak, Multi-Hamiltonian theory of dynamical systems, in: Texts and Monographs in Physics, Springer-Verlag, Berlin, 1998.[7] M. Błaszak, K.Marciniak, From Stäckel systems to integrable hierarchies of PDE’s: Benenti class of separation relations, J. Math. Phys. 47 (2006) 032904.[8] M. Antonowicz, A.P. Fordy, Coupled KdV equations with multi-Hamiltonian structures, Physica D 28 (3) (1987) 345–357.[9] M. Blaszak, K. Marciniak, Stäckel systems generating coupled KdV hierarchies and their finite-gap and rational solutions, J. Phys. A 41 (48) (2008)485202.

[10] M. Antonowicz, A.P. Fordy, Coupled Harry Dym equations with multi-Hamiltonian structures, J. Phys. A 21 (5) (1988) L269–L275.[11] M. Antonowicz, A.P. Fordy, Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems, Comm. Math. Phys. 124 (3)

(1989) 465–486.[12] J.C. Brunelli, G.A.T.F. da Costa, On the nonlocal equations and nonlocal charges associated with the Harry Dym hierarchy, J. Math. Phys. 43 (12) (2002)

6116–6128.[13] John K. Hunter, Yu Xi Zheng, On a completely integrable nonlinear hyperbolic variational equation, Physica D 79 (2-4) (1994) 361–386.[14] E.K. Sklyanin, Separation of variables—new trends, Progr. Theoret. Phys. Suppl. 118 (1995) 35–60.[15] M. Błaszak, Theory of separability of multi-Hamiltonian chains, J. Math. Phys. 40 (1999) 5725.[16] S.P. Tsarëv, Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type, Soviet Math. Dokl. 31 (1985) 488–491.[17] S.P. Tsarëv, The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method, Izv. Akad. Nauk SSSR Ser. Mat. 54 (5)

(1990) 1048–1068 (in Russian); translation in Math. USSR-Izv. 37 (2) (1991) 397–419.[18] E.V. Ferapontov, Integration of weakly nonlinear hydrodynamic ssystems in Riemman invariants, Phys. Lett. A (1991) 112–118.[19] L.M. Alonso, A.B. Shabat, Energy-dependent potentials revisited: a universal hierarchy of hydrodynamic type, Phys. Lett. A 300 (2002) 58–64.

Construction of coupled Harry Dym hierarchy and its solutions from Stäckel systems

Documents

Transcript of Construction of coupled Harry Dym hierarchy and its solutions from Stäckel systems